Lagrangian for Real Systems Instead of Entropy for Ideal Isolated Systems

The Second Law of Thermodynamics states that entropy S increases in a spontaneous process in an ideal isothermal and isolated system. Real systems are influenced by external forces and fields, including the temperature field. In this case, only entropy is not enough, and we suggest using a new function, Ls, which is analogous to the Lagrangian in classical mechanics. It includes total potential energy but instead of mechanical kinetic energy, Ls includes the product ST, and the system always evolves towards increasing this modified Lagrangian. It reaches an equilibrium when total potential force is balanced by both entropic and thermal forces. All forces have the same units, Newton/mol, and may be added or subtracted. For condensed systems with friction forces, it is a molecular transport velocity, and not acceleration, which is proportional to the acting force. Our approach has several advantages compared to Onsager’s non-equilibrium thermodynamics with its thermodynamic forces, which may have different units, including 1/T for energy transport. For isolated systems, the description is reduced to Second Law and Clausius inequality. It easily explains diffusion, Dufour effect, and Soret thermodiffusion. The combination of electric, thermal, and entropic forces explains thermoelectric phenomena, including Peltier–Seebeck and Thomson (Lord Kelvin) effects. Gravitational and entropic forces together inside a black hole may lead to a steady state or the black hole evaporation. They are also involved in and influenced by solar atmospheric processes.